You have found the following ages (in years) of all 6 sloths at your local zoo: $ 3,\enspace 1,\enspace 13,\enspace 1,\enspace 21,\enspace 14$ What is the average age of the sloths at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Answer: Because we have data for all 6 sloths at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$ To find the population mean , add up the values of all $6$ ages and divide by $6$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\mu} = \dfrac{3 + 1 + 13 + 1 + 21 + 14}{{6}} = {8.8\text{ years old}} $ Find the squared deviations from the mean for each sloth. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $3$ years $-5.8$ years $33.64$ years $^2$ $1$ year $-7.8$ years $60.84$ years $^2$ $13$ years $4.2$ years $17.64$ years $^2$ $1$ year $-7.8$ years $60.84$ years $^2$ $21$ years $12.2$ years $148.84$ years $^2$ $14$ years $5.2$ years $27.04$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{33.64} + {60.84} + {17.64} + {60.84} + {148.84} + {27.04}} {{6}} $ $ {\sigma^2} = \dfrac{{348.84}}{{6}} = {58.14\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$ ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{58.14\text{ years}^2}} = {7.6\text{ years}} $ The average sloth at the zoo is 8.8 years old. There is a standard deviation of 7.6 years.